There is a \(T\) -equivariant embedding of \(X\)  into the projective
space \(P(V)\)  for some representation \(V\)  of \(T\) . Given that,
we can assume that \(X=P(V)\) ; this greatly simplifies the situation.
Then \(T\)  acts on \(X=\text{\bf P}^r\)  by
\(t([z_0,\ldots ,z_r])=[t^{a_0}z_0,\ldots ,t^{a_r}z_r]\) 
for some integers \(a_i\) .
We can assume that \(a_0\le \cdots \le a_r\) .
Composing \(f\colon C\rightarrow X\)  with the action of \(T\)  on \(X\) 
gives a morphism \(T\times C\rightarrow X\) , which can be viewed
as a rational map \(G\colon \text{\bf P}^1\times C\dashrightarrow X\)  over \(k\) .
Since \(X\)  is proper over \(k\) ,
\(G\)  becomes a morphism \(W\rightarrow X\) , where \(W\)  is a surface obtained by
blowing up \(\text{\bf P}^1\times C\)  finitely many times
at closed points. In particular, \(G\)  restricts to a morphism \(\text{\bf P}^1\times (C-Z)\rightarrow X\)  for some 0-dimensional closed subset \(Z\)  of \(X\) .
Because \(C\)  is normal and all fibers of \(W\rightarrow C\) 
have dimension 1, the fibers of \(W\rightarrow C\)  form
a well-defined family of effective
1-cycles on \(W\) , and hence they give a morphism from \(C\)  to the Chow
variety of 1-cycles on \(W\)  [1]}. Pushing
cycles forward makes the Chow
variety covariantly functorial under arbitrary morphisms
[1]}. Therefore, the morphism \(W\rightarrow X\) 
gives a morphism \(e\)  from \(C\)  to the Chow variety
of 1-cycles on \(X\) .
For each \(k\) -point \(c\)  in \(C\)  (possibly in \(Z\) ),
\(e(c)\)  is an effective 1-cycle in \(X\)  whose support \(S\)  is the image
under \(G\)  of
the inverse image of \(c\)  in \(W\) . Let us describe this image using
power series.
The completed local ring of \(C\)  at \(c\)  is isomorphic
to the power series ring \(k[[u]]\) . So the curve \(f\colon C\rightarrow X=\text{\bf P}^r\) 
near \(c\)  is given by some power series \([z_0(u),\ldots ,z_r(u)]\) 
with \(z_i(u)\in k[[u]]\) , not all zero.
Every point in the inverse image of \(c\)  in \(W\)  is contained
in some curve in \(W\)  that meets the open set \(T\times (C-Z)\)  in \(W\) .
After completion,
this curve determines a finite extension \(F\)  of the field \(k((u))\) ,
along with an \(F\) -point of \(T\times (C-Z)\)  over the given
\(k((u))\) -point of \(C\) . Therefore,
the support \(S\)  of the limit 1-cycle \(e(c)\) ,
viewed as a subset of \(X(\overline{k})\) ,
is the set of all \(\overline{k}\) -points in \(X\) 
that can be written as
\(p=\lim _{u\rightarrow 0} [g(u)^{a_0}z_0(u),\ldots ,g(u)^{a_r}z_r(u)]\) 
for some \(g\)  in the algebraic closure \(\overline{k((u))}\) , \(g\ne 0\) .
This limit point depends mainly
on the rational number \(b:=\operatorname{ord}_u(g)\) .
The situation is described by the Newton
polygon of the pairs \((a_i,\operatorname{ord}_u(z_i))\)  in \(\text{\bf Z}\times (\text{\bf Z}\cup {\infty })\) ,
as in Figure REF . (Here \(\operatorname{ord}_u(z_i)=\infty \) 
if \(z_i(u)\)  is identically zero.)
<FIGURE>Namely, let \(I\)  be the set of numbers \(i\in \lbrace 0,\ldots ,r\rbrace \)  such that
\(ba_i+\operatorname{ord}_u(z_i)\)  reaches its minimum value. (That is, let \(l\)  be
the unique line of slope \(-b\)  that meets the Newton polygon but not
the region above it; then \(I\)  corresponds to the points
\((a_i,\operatorname{ord}_u(z_i))\)  that lie on \(l\) .)
Then we compute that the limit point \(p\)  defined above has
all coordinates zero except the \(i\) th coordinate for elements \(i\in I\) .
Replacing \(g\)  by another function with the same value of \(b\)  (that is,
multiplying \(g\)  by a unit \(h(u)\) ) just replaces \(p\)  by \(h(0)(p)\) , another point
in the same \(T\) -orbit as \(p\) .
For all but finitely many rational numbers \(b\) ,
the limit point \(p\)  above belongs to a set \(\lbrace x_0,x_1,\ldots ,x_n\rbrace \) 
of \(T\) -fixed points, these being
indexed by the vertices of the Newton polygon. (For these values of \(b\) ,
all nonzero coordinates \(i\)  in the set \(I\)  above have the same weight \(a_i\) ,
which means that \(p\)  is a \(T\) -fixed point.) For the remaining \(n\) 
values of \(b\) ,
corresponding to the non-vertical edges of the Newton polygon,
the limit point can be anywhere in the \(T\) -orbit of a certain point \(y_i\) 
in \(X\) 
with \(\lim _{t\rightarrow 0}t(y_i)=x_{i-1}\)  and \(\lim _{t\rightarrow \infty }t(y_i)=x_i\) .
Here the points \(y_1,\ldots ,y_n\)  (and hence the points \(x_0,x_1,\ldots ,x_n\) ) can be taken to be
\(k\) -points of \(X\) , by choosing the function \(g\in \overline{k((u))}\) 
with a given
value of \(\operatorname{ord}_u(g)\in \text{\bf Q}\)  to lie in a totally ramified extension
of \(k((u))\) , for example in \(k((u^{1/e}))\)  for a positive integer \(e\) .
